# L11n237

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n237 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{\left(t(2) t(1)^2-2 t(1)^2-2 t(2)+1\right) \left(t(1) t(2)^2+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -4 (db) HOMFLY-PT polynomial $2 a^9 z+a^9 z^{-1} -a^7 z^5-6 a^7 z^3-7 a^7 z-a^7 z^{-1} +a^5 z^7+6 a^5 z^5+10 a^5 z^3+6 a^5 z-a^3 z^5-5 a^3 z^3-5 a^3 z$ (db) Kauffman polynomial $-z^5 a^{11}+4 z^3 a^{11}-2 z a^{11}-z^6 a^{10}+4 z^4 a^{10}-2 z^2 a^{10}-z^7 a^9+6 z^5 a^9-10 z^3 a^9+7 z a^9-a^9 z^{-1} -z^6 a^8+6 z^4 a^8-6 z^2 a^8+a^8-2 z^7 a^7+13 z^5 a^7-21 z^3 a^7+9 z a^7-a^7 z^{-1} -z^8 a^6+6 z^6 a^6-8 z^4 a^6+2 z^2 a^6-2 z^7 a^5+12 z^5 a^5-17 z^3 a^5+5 z a^5-z^8 a^4+6 z^6 a^4-10 z^4 a^4+6 z^2 a^4-z^7 a^3+6 z^5 a^3-10 z^3 a^3+5 z a^3$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-8-7-6-5-4-3-2-1012χ
0          11
-2           0
-4       121 0
-6       11  0
-8     221   -1
-10    111    1
-12   131     1
-14  111      1
-16  11       0
-1811         0
-201          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.